Verified and published
Set August 7, 2026
Covering design C(28,13,4) ≤ 53
A covering design is a pack of blocks that together contain every small subset of a set. Fewer blocks doing the same job is a better answer, and the best known answers live in a public repository that mathematicians have been chipping at for decades.
My program found a design that covers everything C(28,13,4) requires with 53 blocks, one under the 54 block record that had stood since September 2008. Dan Gordon, who created the La Jolla Covering Repository, verified it independently with his own code. It is published CC0, free for anyone to use, cite, or try to beat.
Verified, beats the live record
Set August 8, 2026
Covering design C(32,13,4) ≤ 95
One day later the same program went bigger. All 35,960 subsets of four are covered by 95 blocks, one under the 96 block mark that had stood since September 2009. My verification is complete: every subset checked, none missed.
Cross checked against the live repository on August 8, 2026: the table still lists 96. Seventeen years, gone overnight. Submission to the repository is in progress.
Complete verdicts, machine checked
Proved August 7 and 8, 2026
Six proofs of emptiness
Ramsey numbers are famously brutal to pin down, so part of the hunt is proving where the answer is not hiding. Six complete SAT verdicts now close six doors for good: no cyclic Z45, Z3 x Z15 Cayley, block circulant 3x15, cyclic Z55, nonabelian F55 Cayley, or cyclic Z61 edge coloring can improve the classical lower bounds R(3,3,5) ≥ 46, R(3,4,4) ≥ 56, and R(3,3,6) ≥ 62.
A negative like that is still knowledge. Each verdict carries a certificate a machine can check, so nobody ever has to search those rooms again.
All of it is my own program under my own direction, run on my own machine. The facts above are the whole claim: nothing rounded up, nothing dressed up.